# Impact of ξ on gravitational wave

In document Towards a gauge-invariant treatment of the electroweak phase transition (Page 42-47)

[Cheng-Wei Chiang, E.S., 1707.06765 (PLB)]

eff

B-L

### model

no resum ⇠ = 0 ⇠ = 1 ⇠ = 5

↵ 2.27 1.44 0.99 0.48

˜ 89.4 97.5 105.4 135.0

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z 1

0

dr r2

1 2

d S dr

2

+ Ve↵( S; T ) , (15)

where S(r) = p

2hS(r)i. The equation of motion for S is then d2 S

dr2 + 2 r

d S dr

@Ve↵

@ S = 0 , (16)

with the boundary conditions: limr!1 S(r) = 0 and d S(r)/dr|r=0 = 0. We can solve Eq. (16) by use of a relaxation method (see, e.g., Ref.  for details).

Let T be the temperature at which the GWs are produced from the cosmological phase transition. Without significant reheating, this temperature can be approximated by the bubble nucleation temperature, TN, to be defined below. For the phase transition to develop, at least one bubble must nucleate within the Hubble volume. We thus define TN through the condition

N(TN) = H4(TN) , (17)

where H(T ) = 1.66p

g(T )T2/mPl with g(T ) being the relativistic degrees of freedom at T and mPl = 1.22 ⇥ 1019 GeV, while N(T ) is the bubble nucleation rate per unit time per unit volume approximately given by 

N(T ) ' T4

S3(T ) 2⇡T

3/2

e S3(T )/T . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref.  using two parameters:

✏(T)

rad(T) and ⌘ HT d dT

S3(T ) T

T =T

, (19)

where

✏(T ) = Ve↵ T @ Ve↵

@T and ⇢rad(T ) = 2

30g(T )T4, (20) 7

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z 1

0

dr r2

1 2

d S dr

2

+ Ve↵( S; T ) , (15) where S(r) = p

2hS(r)i. The equation of motion for S is then d2 S

dr2 + 2 r

d S dr

@Ve↵

@ S = 0 , (16)

with the boundary conditions: limr!1 S(r) = 0 and d S(r)/dr|r=0 = 0. We can solve Eq. (16) by use of a relaxation method (see, e.g., Ref.  for details).

Let T be the temperature at which the GWs are produced from the cosmological phase transition. Without significant reheating, this temperature can be approximated by the bubble nucleation temperature, TN, to be defined below. For the phase transition to develop, at least one bubble must nucleate within the Hubble volume. We thus define TN through the condition

N(TN) = H4(TN) , (17)

where H(T ) = 1.66p

g(T )T2/mPl with g(T ) being the relativistic degrees of freedom at T and mPl = 1.22 ⇥ 1019 GeV, while N(T ) is the bubble nucleation rate per unit time per unit volume approximately given by 

N(T ) ' T4

S3(T ) 2⇡T

3/2

e S3(T )/T . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref.  using two parameters:

✏(T)

rad(T) and ⌘ HT d dT

S3(T ) T

T =T

, (19)

where

✏(T ) = Ve↵ T @ Ve↵

@T and ⇢rad(T ) = 2

30g(T )T4, (20) 7

10-18 10-16 10-14 10-12 10-10 10-8 10-6

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102

f [Hz]

GWh2

no resum ξ = 0 ξ = 1 ξ = 5

### Impact of ξ on gravitational wave

[Cheng-Wei Chiang, E.S., 1707.06765 (PLB)]

eff

B-L

### ξ =0

no resum ⇠ = 0 ⇠ = 1 ⇠ = 5

↵ 2.27 1.44 0.99 0.48

˜ 89.4 97.5 105.4 135.0

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z 1

0

dr r2

1 2

d S dr

2

+ Ve↵( S; T ) , (15)

where S(r) = p

2hS(r)i. The equation of motion for S is then d2 S

dr2 + 2 r

d S dr

@Ve↵

@ S = 0 , (16)

with the boundary conditions: limr!1 S(r) = 0 and d S(r)/dr|r=0 = 0. We can solve Eq. (16) by use of a relaxation method (see, e.g., Ref.  for details).

Let T be the temperature at which the GWs are produced from the cosmological phase transition. Without significant reheating, this temperature can be approximated by the bubble nucleation temperature, TN, to be defined below. For the phase transition to develop, at least one bubble must nucleate within the Hubble volume. We thus define TN through the condition

N(TN) = H4(TN) , (17)

where H(T ) = 1.66p

g(T )T2/mPl with g(T ) being the relativistic degrees of freedom at T and mPl = 1.22 ⇥ 1019 GeV, while N(T ) is the bubble nucleation rate per unit time per unit volume approximately given by 

N(T ) ' T4

S3(T ) 2⇡T

3/2

e S3(T )/T . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref.  using two parameters:

✏(T)

rad(T) and ⌘ HT d dT

S3(T ) T

T =T

, (19)

where

✏(T ) = Ve↵ T @ Ve↵

@T and ⇢rad(T ) = 2

30g(T )T4, (20) 7

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z 1

0

dr r2

1 2

d S dr

2

+ Ve↵( S; T ) , (15) where S(r) = p

2hS(r)i. The equation of motion for S is then d2 S

dr2 + 2 r

d S dr

@Ve↵

@ S = 0 , (16)

with the boundary conditions: limr!1 S(r) = 0 and d S(r)/dr|r=0 = 0. We can solve Eq. (16) by use of a relaxation method (see, e.g., Ref.  for details).

Let T be the temperature at which the GWs are produced from the cosmological phase transition. Without significant reheating, this temperature can be approximated by the bubble nucleation temperature, TN, to be defined below. For the phase transition to develop, at least one bubble must nucleate within the Hubble volume. We thus define TN through the condition

N(TN) = H4(TN) , (17)

where H(T ) = 1.66p

g(T )T2/mPl with g(T ) being the relativistic degrees of freedom at T and mPl = 1.22 ⇥ 1019 GeV, while N(T ) is the bubble nucleation rate per unit time per unit volume approximately given by 

N(T ) ' T4

S3(T ) 2⇡T

3/2

e S3(T )/T . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref.  using two parameters:

✏(T)

rad(T) and ⌘ HT d dT

S3(T ) T

T =T

, (19)

where

✏(T ) = Ve↵ T @ Ve↵

@T and ⇢rad(T ) = 2

30g(T )T4, (20) 7

10-18 10-16 10-14 10-12 10-10 10-8 10-6

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102

f [Hz]

GWh2

no resum ξ = 0 ξ = 1 ξ = 5

### Impact of ξ on gravitational wave

[Cheng-Wei Chiang, E.S., 1707.06765 (PLB)]

eff

B-L

### ξ =0

no resum ⇠ = 0 ⇠ = 1 ⇠ = 5

↵ 2.27 1.44 0.99 0.48

˜ 89.4 97.5 105.4 135.0

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z 1

0

dr r2

1 2

d S dr

2

+ Ve↵( S; T ) , (15)

where S(r) = p

2hS(r)i. The equation of motion for S is then d2 S

dr2 + 2 r

d S dr

@Ve↵

@ S = 0 , (16)

with the boundary conditions: limr!1 S(r) = 0 and d S(r)/dr|r=0 = 0. We can solve Eq. (16) by use of a relaxation method (see, e.g., Ref.  for details).

Let T be the temperature at which the GWs are produced from the cosmological phase transition. Without significant reheating, this temperature can be approximated by the bubble nucleation temperature, TN, to be defined below. For the phase transition to develop, at least one bubble must nucleate within the Hubble volume. We thus define TN through the condition

N(TN) = H4(TN) , (17)

where H(T ) = 1.66p

g(T )T2/mPl with g(T ) being the relativistic degrees of freedom at T and mPl = 1.22 ⇥ 1019 GeV, while N(T ) is the bubble nucleation rate per unit time per unit volume approximately given by 

N(T ) ' T4

S3(T ) 2⇡T

3/2

e S3(T )/T . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref.  using two parameters:

✏(T)

rad(T) and ⌘ HT d dT

S3(T ) T

T =T

, (19)

where

✏(T ) = Ve↵ T @ Ve↵

@T and ⇢rad(T ) = 2

30g(T )T4, (20) 7

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z 1

0

dr r2

1 2

d S dr

2

+ Ve↵( S; T ) , (15) where S(r) = p

2hS(r)i. The equation of motion for S is then d2 S

dr2 + 2 r

d S dr

@Ve↵

@ S = 0 , (16)

with the boundary conditions: limr!1 S(r) = 0 and d S(r)/dr|r=0 = 0. We can solve Eq. (16) by use of a relaxation method (see, e.g., Ref.  for details).

Let T be the temperature at which the GWs are produced from the cosmological phase transition. Without significant reheating, this temperature can be approximated by the bubble nucleation temperature, TN, to be defined below. For the phase transition to develop, at least one bubble must nucleate within the Hubble volume. We thus define TN through the condition

N(TN) = H4(TN) , (17)

where H(T ) = 1.66p

g(T )T2/mPl with g(T ) being the relativistic degrees of freedom at T and mPl = 1.22 ⇥ 1019 GeV, while N(T ) is the bubble nucleation rate per unit time per unit volume approximately given by 

N(T ) ' T4

S3(T ) 2⇡T

3/2

e S3(T )/T . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref.  using two parameters:

✏(T)

rad(T) and ⌘ HT d dT

S3(T ) T

T =T

, (19)

where

✏(T ) = Ve↵ T @ Ve↵

@T and ⇢rad(T ) = 2

30g(T )T4, (20) 7

10-18 10-16 10-14 10-12 10-10 10-8 10-6

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102

f [Hz]

GWh2

no resum ξ = 0 ξ = 1 ξ = 5

### Impact of ξ on gravitational wave

[Cheng-Wei Chiang, E.S., 1707.06765 (PLB)]

eff

B-L

### ξ =0 ξ =5

no resum ⇠ = 0 ⇠ = 1 ⇠ = 5

↵ 2.27 1.44 0.99 0.48

˜ 89.4 97.5 105.4 135.0

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z 1

0

dr r2

1 2

d S dr

2

+ Ve↵( S; T ) , (15)

where S(r) = p

2hS(r)i. The equation of motion for S is then d2 S

dr2 + 2 r

d S dr

@Ve↵

@ S = 0 , (16)

with the boundary conditions: limr!1 S(r) = 0 and d S(r)/dr|r=0 = 0. We can solve Eq. (16) by use of a relaxation method (see, e.g., Ref.  for details).

Let T be the temperature at which the GWs are produced from the cosmological phase transition. Without significant reheating, this temperature can be approximated by the bubble nucleation temperature, TN, to be defined below. For the phase transition to develop, at least one bubble must nucleate within the Hubble volume. We thus define TN through the condition

N(TN) = H4(TN) , (17)

where H(T ) = 1.66p

g(T )T2/mPl with g(T ) being the relativistic degrees of freedom at T and mPl = 1.22 ⇥ 1019 GeV, while N(T ) is the bubble nucleation rate per unit time per unit volume approximately given by 

N(T ) ' T4

S3(T ) 2⇡T

3/2

e S3(T )/T . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref.  using two parameters:

✏(T)

rad(T) and ⌘ HT d dT

S3(T ) T

T =T

, (19)

where

✏(T ) = Ve↵ T @ Ve↵

@T and ⇢rad(T ) = 2

30g(T )T4, (20) 7

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z 1

0

dr r2

1 2

d S dr

2

+ Ve↵( S; T ) , (15) where S(r) = p

2hS(r)i. The equation of motion for S is then d2 S

dr2 + 2 r

d S dr

@Ve↵

@ S = 0 , (16)

with the boundary conditions: limr!1 S(r) = 0 and d S(r)/dr|r=0 = 0. We can solve Eq. (16) by use of a relaxation method (see, e.g., Ref.  for details).

Let T be the temperature at which the GWs are produced from the cosmological phase transition. Without significant reheating, this temperature can be approximated by the bubble nucleation temperature, TN, to be defined below. For the phase transition to develop, at least one bubble must nucleate within the Hubble volume. We thus define TN through the condition

N(TN) = H4(TN) , (17)

where H(T ) = 1.66p

g(T )T2/mPl with g(T ) being the relativistic degrees of freedom at T and mPl = 1.22 ⇥ 1019 GeV, while N(T ) is the bubble nucleation rate per unit time per unit volume approximately given by 

N(T ) ' T4

S3(T ) 2⇡T

3/2

e S3(T )/T . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref.  using two parameters:

✏(T)

rad(T) and ⌘ HT d dT

S3(T ) T

T =T

, (19)

where

✏(T ) = Ve↵ T @ Ve↵

@T and ⇢rad(T ) = 2

30g(T )T4, (20) 7

10-18 10-16 10-14 10-12 10-10 10-8 10-6

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102

f [Hz]

GWh2

no resum ξ = 0 ξ = 1 ξ = 5

### Impact of ξ on gravitational wave

[Cheng-Wei Chiang, E.S., 1707.06765 (PLB)]

eff

B-L

### ξ =0 ξ =5

no resum ⇠ = 0 ⇠ = 1 ⇠ = 5

↵ 2.27 1.44 0.99 0.48

˜ 89.4 97.5 105.4 135.0

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z 1

0

dr r2

1 2

d S dr

2

+ Ve↵( S; T ) , (15)

where S(r) = p

2hS(r)i. The equation of motion for S is then d2 S

dr2 + 2 r

d S dr

@Ve↵

@ S = 0 , (16)

with the boundary conditions: limr!1 S(r) = 0 and d S(r)/dr|r=0 = 0. We can solve Eq. (16) by use of a relaxation method (see, e.g., Ref.  for details).

Let T be the temperature at which the GWs are produced from the cosmological phase transition. Without significant reheating, this temperature can be approximated by the bubble nucleation temperature, TN, to be defined below. For the phase transition to develop, at least one bubble must nucleate within the Hubble volume. We thus define TN through the condition

N(TN) = H4(TN) , (17)

where H(T ) = 1.66p

g(T )T2/mPl with g(T ) being the relativistic degrees of freedom at T and mPl = 1.22 ⇥ 1019 GeV, while N(T ) is the bubble nucleation rate per unit time per unit volume approximately given by 

N(T ) ' T4

S3(T ) 2⇡T

3/2

e S3(T )/T . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref.  using two parameters:

✏(T)

rad(T) and ⌘ HT d dT

S3(T ) T

T =T

, (19)

where

✏(T ) = Ve↵ T @ Ve↵

@T and ⇢rad(T ) = 2

30g(T )T4, (20) 7

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z 1

0

dr r2

1 2

d S dr

2

+ Ve↵( S; T ) , (15) where S(r) = p

2hS(r)i. The equation of motion for S is then d2 S

dr2 + 2 r

d S dr

@Ve↵

@ S = 0 , (16)

with the boundary conditions: limr!1 S(r) = 0 and d S(r)/dr|r=0 = 0. We can solve Eq. (16) by use of a relaxation method (see, e.g., Ref.  for details).

Let T be the temperature at which the GWs are produced from the cosmological phase transition. Without significant reheating, this temperature can be approximated by the bubble nucleation temperature, TN, to be defined below. For the phase transition to develop, at least one bubble must nucleate within the Hubble volume. We thus define TN through the condition

N(TN) = H4(TN) , (17)

where H(T ) = 1.66p

g(T )T2/mPl with g(T ) being the relativistic degrees of freedom at T and mPl = 1.22 ⇥ 1019 GeV, while N(T ) is the bubble nucleation rate per unit time per unit volume approximately given by 

N(T ) ' T4

S3(T ) 2⇡T

3/2

e S3(T )/T . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref.  using two parameters:

✏(T)

rad(T) and ⌘ HT d dT

S3(T ) T

T =T

, (19)

where

✏(T ) = Ve↵ T @ Ve↵

@T and ⇢rad(T ) = 2

30g(T )T4, (20) 7

In document Towards a gauge-invariant treatment of the electroweak phase transition (Page 42-47)